Elements of Effective Math Instruction
Math learning happens best in a robust environment that attends to many needs of the learner. It is more than just learning the concepts, practicing and testing.
The foundation for teaching is the connection we have with each student. Teaching is an extremely personal and is predicated on relationships. Until we establish a connection and build trust, students may not be open to learning. This point has been driven home to me as I now have the opportunity to experience many classrooms. Students are sometimes completely closed to the notion of learning, attempting, and taking risks. Somehow we have to get them to trust us, to understand that we have high expectations for them, we believe in them, they can trust us to help them achieve. We need to explicitly teach students what learning looks like; that learning in mathematics requires taking risks and making mistakes. If no one made any mistakes no one would learn anything. Mistakes and misconceptions are welcome in math class! We can focus our dialogue and interactions around learning and improving, helping students realize that learning looks different for everyone in the room, and that in our class, everyone will grow.
Anyone who has played or coached sports knows the importance of having a clear goal in mind for any practice or training. Every lesson in the classroom needs a clear focus that is communicated to students. Knowing exactly what learning needs to take place keeps the teacher on target and focussed, and also is a guide to students. The purpose (goal, target, outcome) of the lesson is the instrument against which we measure all activity in the classroom. We use it to evaluate what lesson activities (videos, opening activities, discussions) are useful, helps focus group work (teaching students explicitly what group learning is for, holding students accountable for meaningful learning, and having students continually self-monitor their collaborative behaviour). The goal of the lesson comes from meaningful planning, knowing our curriculum, deciding what is important, knowing what success looks like and communicating that to students.
We need to help students take ownership of the learning by constructing meaning for themselves through dialogue, rich tasks, writing and multimedia. This means interacting with the content in different ways. For example, learning to graph linear equations is concept then procedure, but can be taught through exploration using graphing software or graphing calculators, having students generate the relationships between slopes and intercepts based on their observations. Later we ask them to communicate their learning to each other and to us through explanations and writing, rather than just performing the procedure. There is abundant research about the importance of having students explain their reasoning rather than just performing the task.
Using multiple representations deepens mathematical understanding. Opportunities to communicate reasoning help students to understand which representations are efficient and meaningful in a given setting. We are asking our students not only to share strategies but to then understand how to employ the most efficient strategies, both for communicating mathematically and calculating, solving, and proving.
Graphic organizers are proven to be an effective way of helping students construct meaning. There are many types: Mind maps, frayer models, concept attainment activities, identifying similarities and differences, creating posters and diagrams. They are effective at the beginning of a unit or lesson as review strategies (activating prior knowledge), during learning (for identifying relationships or structuring learning) or as summaries (unit review, making connections between topics). These are just a few ideas!
The three “legs of the stool” of a firm foundation in mathematics are: Conceptual learning, procedural fluency, and problem solving. Each of these three is equally important! We still need to have practice and establish automaticity and efficiency. Practice without conceptual understanding is meaningless if we are trying to create mathematically literate students that can apply and transfer their understanding. Too often we neglect what has always been good learning in the past: direct instruction (which still has a significant effect size, especially for higher level procedural topics), and independent practice. But all things in moderation. A lesson can begin with whole group direct instruction (more in higher grades) and then we can also allow opportunities for collaborative learning and small group instruction, and then use activities such as graphic organizers to revisit the learning and retrieve and construct understanding.
Research from neuroscience supports the importance of writing about learning and using spaced practice as ways to solidify learning. Formative assessment, which helps us plan and respond to learning needs and also helps students understand what they know and what they still need to know, can also help provide opportunities for spaced practice. Everything we teach is assessed. The assessment activities are part of the learning, not isolated events that take time away from teaching and learning. This is why we sometimes say “assessment as learning”.
Finally, goal setting and self-monitoring help students achieve self-efficacy, which is a major predictor of success in mathematics. Having students set learning goals for themselves is very motivating, and our journaling and formative assessment practices help students gauge their progress toward those goals . The only way for students to set goals and measure their progress is to involve them in assessment. We need to make assessment criteria very visible to them, even involve them in establishing the criteria. It must be crystal clear what success in our class will look like. A student can hit any target that they can see and that doesn’t move (Anne Davies). This means that in every outcome there will be some dialogue and constructing assessment criteria, and having that criteria posted in the room or copied in student notebooks.
There are so many other things to consider: scaffolding, differentiating, enriching, summarizing and notetaking, explicitly teaching study skills, high level questioning, strategies for struggling math learners, vocabulary strategies, to name a few. Please consider following me on twitter and/or google plus where I continually post research, tips, and resources to address all these areas (I rarely duplicate twitter posts on google plus, its usually one or the other). There is so much available now to teachers and so much expected, the task seems (well let’s face it, the task is) overwhelming. But honour what you have always done, evaluate your instruction, attend to expanding your repertoire and collection of resources, explore new methods and ideas, and above all reflect! I wish you a year of growth and new horizons!